Theorem carden2b 8419

Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 8418 are meant to replace carden 8994 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
carden2b	|- ( A ~~ B -> ( card ` A ) = ( card ` B ) )

Proof of Theorem carden2b

Step	Hyp	Ref	Expression
1		cardne 8417	. . . . 5 |- ( ( card ` B ) e. ( card ` A ) -> -. ( card ` B ) ~~ A )
2		ennum 8399	. . . . . . . 8 |- ( A ~~ B -> ( A e. dom card <-> B e. dom card ) )
3	2	biimpa 492	. . . . . . 7 |- ( ( A ~~ B /\ A e. dom card ) -> B e. dom card )
4		cardid2 8405	. . . . . . 7 |- ( B e. dom card -> ( card ` B ) ~~ B )
5	3, 4	syl 17	. . . . . 6 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) ~~ B )
6		ensym 7636	. . . . . . 7 |- ( A ~~ B -> B ~~ A )
7	6	adantr 472	. . . . . 6 |- ( ( A ~~ B /\ A e. dom card ) -> B ~~ A )
8		entr 7639	. . . . . 6 |- ( ( ( card ` B ) ~~ B /\ B ~~ A ) -> ( card ` B ) ~~ A )
9	5, 7, 8	syl2anc 673	. . . . 5 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) ~~ A )
10	1, 9	nsyl3 123	. . . 4 |- ( ( A ~~ B /\ A e. dom card ) -> -. ( card ` B ) e. ( card ` A ) )
11		cardon 8396	. . . . 5 |- ( card ` A ) e. On
12		cardon 8396	. . . . 5 |- ( card ` B ) e. On
13		ontri1 5464	. . . . 5 |- ( ( ( card ` A ) e. On /\ ( card ` B ) e. On ) -> ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) ) )
14	11, 12, 13	mp2an 686	. . . 4 |- ( ( card ` A ) C_ ( card ` B ) <-> -. ( card ` B ) e. ( card ` A ) )
15	10, 14	sylibr 217	. . 3 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) C_ ( card ` B ) )
16		cardne 8417	. . . . 5 |- ( ( card ` A ) e. ( card ` B ) -> -. ( card ` A ) ~~ B )
17		cardid2 8405	. . . . . 6 |- ( A e. dom card -> ( card ` A ) ~~ A )
18		id 22	. . . . . 6 |- ( A ~~ B -> A ~~ B )
19		entr 7639	. . . . . 6 |- ( ( ( card ` A ) ~~ A /\ A ~~ B ) -> ( card ` A ) ~~ B )
20	17, 18, 19	syl2anr 486	. . . . 5 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) ~~ B )
21	16, 20	nsyl3 123	. . . 4 |- ( ( A ~~ B /\ A e. dom card ) -> -. ( card ` A ) e. ( card ` B ) )
22		ontri1 5464	. . . . 5 |- ( ( ( card ` B ) e. On /\ ( card ` A ) e. On ) -> ( ( card ` B ) C_ ( card ` A ) <-> -. ( card ` A ) e. ( card ` B ) ) )
23	12, 11, 22	mp2an 686	. . . 4 |- ( ( card ` B ) C_ ( card ` A ) <-> -. ( card ` A ) e. ( card ` B ) )
24	21, 23	sylibr 217	. . 3 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` B ) C_ ( card ` A ) )
25	15, 24	eqssd 3435	. 2 |- ( ( A ~~ B /\ A e. dom card ) -> ( card ` A ) = ( card ` B ) )
26		ndmfv 5903	. . . 4 |- ( -. A e. dom card -> ( card ` A ) = (/) )
27	26	adantl 473	. . 3 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` A ) = (/) )
28	2	notbid 301	. . . . 5 |- ( A ~~ B -> ( -. A e. dom card <-> -. B e. dom card ) )
29	28	biimpa 492	. . . 4 |- ( ( A ~~ B /\ -. A e. dom card ) -> -. B e. dom card )
30		ndmfv 5903	. . . 4 |- ( -. B e. dom card -> ( card ` B ) = (/) )
31	29, 30	syl 17	. . 3 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` B ) = (/) )
32	27, 31	eqtr4d 2508	. 2 |- ( ( A ~~ B /\ -. A e. dom card ) -> ( card ` A ) = ( card ` B ) )
33	25, 32	pm2.61dan 808	1 |- ( A ~~ B -> ( card ` A ) = ( card ` B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   C_ wss 3390   (/)c0 3722   class class class wbr 4395   dom cdm 4839   Oncon0 5430   ` cfv 5589   ~~ cen 7584   cardccrd 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-er 7381  df-en 7588  df-card 8391

This theorem is referenced by:  card1  8420  carddom2  8429  cardennn  8435  cardsucinf  8436  pm54.43lem  8451  nnacda  8649  ficardun  8650  ackbij1lem5  8672  ackbij1lem8  8675  ackbij1lem9  8676  ackbij2lem2  8688  carden  8994  r1tskina  9225  cardfz  12221